A new approach to spatial data interpolation using higher-order statistics

Abstract

Interpolation techniques for spatial data have been applied frequently in various fields of geosciences. Although most conventional interpolation methods assume that it is sufficient to use first- and second-order statistics to characterize random fields, researchers have now realized that these methods cannot always provide reliable interpolation results, since geological and environmental phenomena tend to be very complex, presenting non-Gaussian distribution and/or non-linear inter-variable relationship. This paper proposes a new approach to the interpolation of spatial data, which can be applied with great flexibility. Suitable cross-variable higher-order spatial statistics are developed to measure the spatial relationship between the random variable at an unsampled location and those in its neighbourhood. Given the computed cross-variable higher-order spatial statistics, the conditional probability density function is approximated via polynomial expansions, which is then utilized to determine the interpolated value at the unsampled location as an expectation. In addition, the uncertainty associated with the interpolation is quantified by constructing prediction intervals of interpolated values. The proposed method is applied to a mineral deposit dataset, and the results demonstrate that it outperforms kriging methods in uncertainty quantification. The introduction of the cross-variable higher-order spatial statistics noticeably improves the quality of the interpolation since it enriches the information that can be extracted from the observed data, and this benefit is substantial when working with data that are sparse or have non-trivial dependence structures.

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Notes

  1. 1.

    Due to confidentiality issues, drill-hole samples are uniformly coloured in the map, i.e. specific values of Cu and S are not displayed.

  2. 2.

    As pointed out by Li et al. (2010), ordinary kriging is the most widely used geostatistical method, producing the best linear unbiased predictions.

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Acknowledgments

The authors would like to acknowledge the support of CRC-ORE, established and supported by the Australian Government’s Cooperative Research Centres Programme. The authors are also grateful to the two anonymous referees for their valuable comments and suggestions that have contributed to improving the quality and presentation of this paper.

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Correspondence to Erhan Kozan.

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Liu, S., Anh, V., McGree, J. et al. A new approach to spatial data interpolation using higher-order statistics. Stoch Environ Res Risk Assess 29, 1679–1690 (2015). https://doi.org/10.1007/s00477-014-0985-1

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Keywords

  • Geostatistics
  • Interpolation
  • Uncertainty quantification
  • Mineral deposit